91Ó°ÊÓ

When we talk about fourth roots, we are referring to a number that, when raised to the power of 4, equals the original number inside the radical. In simple terms, if we have \( \root[4]{a} \), it means finding the number \( b \) so that \( b^4 = a \).
For example, the fourth root of 16 is 2 because \( 2^4 = 16 \).
Fourth roots simplify numbers and variables that have exponents which are multiples of 4. This is a key step in simplifying more complex expressions involving radicals. In the problem given, the number 16 can be directly written as \( 2^4 \), making it easier to find the fourth root.

Exponents tell us how many times a number is multiplied by itself. For example, \( x^5 \) means \( x \cdot x \cdot x \cdot x \cdot x \). Exponents are crucial when dealing with radicals, especially higher-order roots like fourth roots.
In the given exercise, the variable \( x^5 \) can be rewritten to simplify the fourth root. Notice that \( x^5 \) can be split into \( x^4 \cdot x \). Because we're dealing with the fourth root, we separate \( x^4 \) (which is a perfect fourth power) from the remaining \( x \). Thus, \( \root[4]{x^5} = x \sqrt[4]{x} \). This principle is central to simplifying all higher-order radicals.

Radicals are symbols that indicate the root of a number. The most common radical is the square root (\( \sqrt{} \)), but they can represent higher roots, like cube roots (\( \sqrt[3]{} \)) or fourth roots (\( \sqrt[4]{} \)). In our problem, we are using the fourth root.
To simplify a radical expression, find and simplify the components inside the radical. This often involves factoring the number or variable into a form where the root can be easily identified. For example, \( \sqrt[4]{16 \cdot x^5 \cdot y^{11}} \) was broken down into simpler parts: \( \sqrt[4]{16} \), \( \sqrt[4]{x^5} \), and \( \sqrt[4]{y^{11}} \), allowing us to simplify each part individually.

Simplifying variables within radicals involves using properties of exponents. When simplifying an expression like \( \sqrt[4]{x^5} \), you break it down into smaller parts that match the radical's order. In our case, we noted that \( x^5 \) is the same as \( x^4 \cdot x \). The \( x^4 \) part is a perfect fourth power, so \( \sqrt[4]{x^4} = x \) and we are left with \( \sqrt[4]{x} \) inside the radical.
Applying the same logic, for \( y^{11} \), it was simplified to \( y^8 \cdot y^3 \). The \( y^8 \) part is a perfect fourth power \( (y^2)^4 \), so its fourth root is \( y^2 \), and we are left with \( \sqrt[4]{y^3} \). This method simplifies the expression step by step, making it easier to handle.